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A306929 Numbers k (>0) such that x^2+y^2 and x^2+k*y^2 can be simultaneously squares.

%I A306929 #29 Oct 03 2022 16:41:25
%S A306929 1,7,10,11,17,20,22,23,24,27,30,31,34,41,42,45,47,49,50,52,53,57,58,
%T A306929 59,60,61,68,71,72,74,76,77,79,82,83,85,86,90,92,93,94,97,99,100,101,
%U A306929 102,104,105,107,110,111,112,113,114,115,119,120,121,122,124,126,127,130,133,134,137
%N A306929 Numbers k (>0) such that x^2+y^2 and x^2+k*y^2 can be simultaneously squares.
%C A306929 Note that Dickson refers to C. H. Brooks and S. Watson, 1857 and lists "the following 41 positive integers A<=100." but 47, 53 and 83 are missing. - _Michael Somos_, Feb 09 2020
%D A306929 The Lady's and Gentleman's Diary, London, 1857, pp. 61-63. See question 1911.
%H A306929 Seiichi Manyama, <a href="/A306929/b306929.txt">Table of n, a(n) for n = 1..1000</a>
%H A306929 L. E. Dickson, <a href="https://archive.org/details/historyoftheoryo02dickuoft/page/474/mode/2up">History of the Theory of Numbers, vol. 2</a>. Carnegie Institute Public. 256, Washington, DC, 1920, see <a href="https://archive.org/details/historyoftheoryo02dickuoft/page/474/mode/2up">p. 475</a>.
%H A306929 H. Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/ec/eca1/cords.txt">Tables of Representation of concordant number n</a>
%H A306929 K. S. Brown, <a href="https://www.mathpages.com/home/kmath286.htm">Concordant Forms</a>
%e A306929 From _Seiichi Manyama_, Jul 15 2019: (Start)
%e A306929 14663^2 + 111384^2 = 112345^2 and 14663^2 + 47*111384^2 = 763751^2. So 47 is a term.
%e A306929 2873161^2 + 2401080^2 = 3744361^2 and 2873161^2 + 83*2401080^2 = 22062761^2. So 83 is a term. (End)
%e A306929 From the K. S. Brown link, 1141^2 + 13260^2 = 13309^2, 1141^2 + 53*13260^2 = 96541^2, so 53 is a term. - _Michael Somos_, Feb 10 2020
%K A306929 nonn
%O A306929 1,2
%A A306929 _N. J. A. Sloane_, Mar 16 2019
%E A306929 More terms from _Seiichi Manyama_, Jul 15 2019